Question

If cos A = 1/3 with A in QIV, then sin A/2=

Answer 1

The question gives us the value of

$\cos A=\frac{1}{3}$

We are then required to find

$\sin (\frac{A}{2})$

In order to find the value of this expression, we need to use the following trigonometric identity:

$\begin{gathered} \cos A=\cos ^2(\frac{A}{2})-\sin ^2(\frac{A}{2}) \\ \\ \cos ^2(\frac{A}{2})=1-\sin ^2(\frac{A}{2}) \\ \\ \therefore\cos A=1-\sin ^2(\frac{A}{2})-\sin ^2(\frac{A}{2}) \\ \cos A=1-2\sin ^2(\frac{A}{2}) \end{gathered}$

With this derived identity for cos A, we can proceed to solve the question.

The identity expresses cos A in terms of sin (A/2). Since we already know the value for cos A, we can proceed to find

the value of sin(A/2)

This is done below:

$\begin{gathered} \cos A=1-2\sin ^2(\frac{A}{2}) \\ \\ \text{Making sin(}\frac{A}{2})\text{ the subject of the formula;} \\ \text{subtract 1 from both sides}S \\ \cos A-1=-2\sin ^2(\frac{A}{2}) \\ \\ \text{Divide both sides by -2} \\ \frac{\cos A-1}{-2}=\sin ^2(\frac{A}{2}) \\ \\ \text{ Find the square root of both sides} \\ \\ \therefore\sin (\frac{A}{2})=\sqrt[]{\frac{\cos A-1}{-2}} \end{gathered}$

Now that we have the final expression for calculating sin(A/2), let us substitute the value of cos A into the expression.

This is done below:

$\begin{gathered} \sin (\frac{A}{2})=\sqrt[]{\frac{\cos A-1}{-2}} \\ \cos A=\frac{1}{3} \\ \\ \sin (\frac{A}{2})=\sqrt[]{\frac{\frac{1}{3}-1}{-2}} \\ \\ \sin (\frac{A}{2})=\sqrt[]{\frac{1}{3}} \\ \\ \end{gathered}$